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DrawMorseLink

PD Presentation:

X6172 X2536 X18,12,19,11 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X20,16,21,15 X22,18,13,17 X16,22,17,21 X12,20,9,19

Gauss Code:

{{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, {7, -6, 8, -10, 9, -3, 11, -8, 10, -9}}

Jones Polynomial:

- q-11/2 + q-9/2 - 6q-7/2 + 7q-5/2 - 16q-3/2 + 16q-1/2 - 19q1/2 + 17q3/2 - 14q5/2 + 10q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + 3q-16 + 5q-14 + 11q-12 + 15q-10 + 15q-8 + 21q-6 + 13q-4 + 10q-2 + 6 - 4q2 - 8q6 - 2q8 - 2q10 - 4q12 + 2q14 - q16

HOMFLY-PT Polynomial:

a-3z-3 + 3a-3z-1 + 3a-3z + 2a-3z3 + a-3z5 - 5a-1z-3 - 14a-1z-1 - 15a-1z - 9a-1z3 - 4a-1z5 - a-1z7 + 9az-3 + 22az-1 + 22az + 12az3 + 3az5 - 7a3z-3 - 14a3z-1 - 11a3z - 3a3z3 + 2a5z-3 + 3a5z-1 + a5z

Kauffman Polynomial:

- a-6z4 - 4a-5z5 - a-4z-2 + 4a-4 - 6a-4z2 + 10a-4z4 - 10a-4z6 + a-3z-3 - 2a-3z-1 + 2a-3z - 8a-3z3 + 20a-3z5 - 14a-3z7 - 7a-2z-2 + 24a-2 - 34a-2z2 + 19a-2z4 + 9a-2z6 - 11a-2z8 + 5a-1z-3 - 12a-1z-1 + 16a-1z - 22a-1z3 + 28a-1z5 - 6a-1z7 - 5a-1z9 - 18z-2 + 58 - 75z2 + 34z4 + 16z6 - 11z8 - z10 + 9az-3 - 24az-1 + 37az - 42az3 + 21az5 + 6az7 - 6az9 - 19a2z-2 + 60a2 - 73a2z2 + 36a2z4 - 2a2z6 - a2z8 - a2z10 + 7a3z-3 - 23a3z-1 + 39a3z - 42a3z3 + 23a3z5 - 3a3z7 - a3z9 - 7a4z-2 + 23a4 - 26a4z2 + 10a4z4 + a4z6 - a4z8 + 2a5z-3 - 9a5z-1 + 16a5z - 14a5z3 + 6a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                       1 j = 10                     3   j = 8                   7 1   j = 6                 7 3     j = 4               10 7       j = 2             9 7         j = 0           11 14           j = -2         5 5             j = -4       2 11               j = -6     4 5                 j = -8   1 6                   j = -10                         j = -12 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	4
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 532]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 532]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[18, 12, 19, 11], X[10, 3, 11, 4], > X[4, 9, 1, 10], X[14, 7, 15, 8], X[8, 13, 5, 14], X[20, 16, 21, 15], > X[22, 18, 13, 17], X[16, 22, 17, 21], X[12, 20, 9, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, > {7, -6, 8, -10, 9, -3, 11, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) -(9/2) 6 7 16 16 3/2 -q + q - ---- + ---- - ---- + ------- - 19 Sqrt[q] + 17 q - 7/2 5/2 3/2 Sqrt[q] q q q 5/2 7/2 9/2 11/2 > 14 q + 10 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 3 5 11 15 15 21 13 10 2 6 8 6 + q + --- + --- + --- + --- + -- + -- + -- + -- - 4 q - 8 q - 2 q - 16 14 12 10 8 6 4 2 q q q q q q q q 10 12 14 16 > 2 q - 4 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 532]][a, z]
Out[8]=	3 5 3 5 1 5 9 a 7 a 2 a 3 14 22 a 14 a 3 a 3 z ----- - ---- + --- - ---- + ---- + ---- - --- + ---- - ----- + ---- + --- - 3 3 3 3 3 3 3 a z z z z 3 a z a z z z z a z a 3 3 5 15 z 3 5 2 z 9 z 3 3 3 z > ---- + 22 a z - 11 a z + a z + ---- - ---- + 12 a z - 3 a z + -- - a 3 a 3 a a 5 7 4 z 5 z > ---- + 3 a z - -- a a
In[9]:=	Kauffman[Link[11, Alternating, 532]][a, z]
Out[9]=	3 5 4 24 2 4 1 5 9 a 7 a 2 a 18 1 58 + -- + -- + 60 a + 23 a + ----- + ---- + --- + ---- + ---- - -- - ----- - 4 2 3 3 3 3 3 3 2 4 2 a a a z a z z z z z a z 2 4 3 5 7 19 a 7 a 2 12 24 a 23 a 9 a 2 z 16 z > ----- - ----- - ---- - ---- - --- - ---- - ----- - ---- + --- + ---- + 2 2 2 2 3 a z z z z 3 a a z z z a z a 2 2 3 5 2 6 z 34 z 2 2 4 2 > 37 a z + 39 a z + 16 a z - 75 z - ---- - ----- - 73 a z - 26 a z - 4 2 a a 3 3 4 4 4 8 z 22 z 3 3 3 5 3 4 z 10 z 19 z > ---- - ----- - 42 a z - 42 a z - 14 a z + 34 z - -- + ----- + ----- + 3 a 6 4 2 a a a a 5 5 5 2 4 4 4 4 z 20 z 28 z 5 3 5 5 5 > 36 a z + 10 a z - ---- + ----- + ----- + 21 a z + 23 a z + 6 a z + 5 3 a a a 6 6 7 7 6 10 z 9 z 2 6 4 6 14 z 6 z 7 3 7 > 16 z - ----- + ---- - 2 a z + a z - ----- - ---- + 6 a z - 3 a z - 4 2 3 a a a a 8 9 5 7 8 11 z 2 8 4 8 5 z 9 3 9 10 2 10 > a z - 11 z - ----- - a z - a z - ---- - 6 a z - a z - z - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 6 4 5 2 11 5 14 + 9 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 8 5 8 4 6 4 6 3 4 3 4 2 2 2 q t q t q t q t q t q t q t q t 11 5 2 4 4 2 6 2 6 3 8 3 > -- + ---- + 7 q t + 10 q t + 7 q t + 7 q t + 3 q t + 7 q t + t 2 q t 8 4 10 4 12 5 > q t + 3 q t + q t

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